Solution
= Solution
Use the tunnel frame $S$, with the entrance at $x=0$, the far end at $x=L$, and $t=0$ at both runner entry and photon emission. The runner has $x_R=vt$, while the approaching <photon> has $x_\gamma=L-ct$. At detection,
$$
t_D=\frac{L}{c+v},\qquad x_D=\frac{vL}{c+v},
\qquad
\boxed{f=\frac{x_D}{L}=\frac{v}{c+v}=\frac{\beta}{1+\beta},\quad\beta=\frac vc.}
$$
For $0<v<c$, $0<f<1/2$: the photon meets the runner before the runner reaches the tunnel midpoint.